Coupled solutions for a bivariate weakly nonexpansive operator by iterations
© Berinde et al.; licensee Springer. 2014
Received: 12 April 2014
Accepted: 30 June 2014
Published: 22 July 2014
We prove weak and strong convergence theorems for a double Krasnoselskij-type iterative method to approximate coupled solutions of a bivariate nonexpansive operator , where C is a nonempty closed and convex subset of a Hilbert space. The new convergence theorems generalize, extend, improve, and complement very important old and recent results in coupled fixed point theory. Some appropriate examples to illustrate our new results and their generalization are also given.
1 Introduction and preliminaries
The study of coupled fixed points has been considered in 2006 by Bhaskar and Lakshmikantham  (see also ). A rich literature on the existence of coupled fixed points of mixed monotone, monotone and non-monotone mappings, has been developed ever since the publication of that paper (see [3–30]).
In particular, the authors establish three kinds of coupled fixed point results: (1) existence theorems (Theorems 2.1 and 2.2); (2) an existence and uniqueness theorem (Theorem 2.4); and (3) theorems that ensure the equality of the coupled fixed point components (Theorems 2.5 and 2.6).
Theorem 1 (, Theorem 2.1 and Theorem 2.6)
Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on X.
Suppose, additionally, that are comparable. Then for the coupled fixed point , we have .
Contraction-type conditions arise naturally in connection with Lipshitzian properties of mappings in the study of nonlinear functional differential and integral equations. Therefore, coupled fixed point results for contractions have important applications in nonlinear analysis and have been applied successfully for solving various classes of nonlinear functional equations: integral equations and systems of integral equations [5, 7, 13, 17, 18, 26, 27, 31]; (periodic) two point boundary value problems [1, 10, 15, 28]; nonlinear Hammerstein integral equations ; nonlinear elliptic problems and delayed hematopoesis models ; systems of differential and integral equations ; nonlinear matrix and nonlinear quadratic equations [4, 13], initial value problems for ODE [6, 24], etc.
We note that in all the above mentioned cases, the main conclusion is drawn using (, Theorem 2.6), which guarantees existence as well as equality of components of the coupled fixed point.
On the other hand, in almost all the papers dealing with study of coupled fixed points, no attention is paid to the constructive features of such a result, i.e., there is neither explicit mention of the method by which one could approximate that coupled fixed point, nor on the order of convergence and/or error estimates of the iteration processes involved.
Moreover, there exist (mixed) monotone mappings (see Examples 1 and 3 below), which possess coupled fixed points, for which no coupled fixed point theorem existing in literature can be applied. This is mainly because all those theorems (we refer here only to the ones given in [1, 4–7, 10, 13, 15–18, 23–31]) are based on a strict contractive-type condition (1.1).
All the above observations motivate for constructive study of coupled fixed points of a bivariate mapping satisfying a weaker contractive condition of nonexpansiveness type and providing a constructive method to approximate these coupled fixed points, which we generally meet in applications, i.e., when we have equality of the coupled fixed point components.
The only paper that considers asymptotically nonexpansive bivariate mappings and the existence of their coupled fixed points is due to Olaoluwa et al. . No other attempt has been made to tackle this important problem. We find here coupled solutions for a bivariate weakly nonexpansive operator on Hilbert spaces through an iterative method. Since nonexpansive bivariate mappings are particular sub-classes of the weakly nonexpansive mappings considered in the present paper, our results also generalize, improve and complement the corresponding results obtained in .
In order to illustrate the broader scope and novelty of our results, we present appropriate examples to delineate them from the existing coupled fixed point theorems in literature and indicate their potential use in applications.
2 Nonexpansive bivariate operators
In this paper, we define the concept of nonexpansiveness for bivariate mappings as follows.
for all , where and .
A similar but stronger concept has been introduced in .
Definition 2 ()
for all .
Note that our condition (2.1) is more general than (2.2): any nonexpansive mapping F is weakly nonexpansive but the converse is not true, in general, as shown below.
Then F satisfies condition (2.1) but does not satisfy condition (2.2). Moreover, F possesses a unique coupled fixed point of the form , i.e., , but no coupled fixed point theorem established in [1, 4–7, 10, 13, 15–18, 23–31] (and in other related papers) can be applied to this function F.
First, let us note that (2.1) holds with the constants and . Suppose F satisfies (2.2).
Then, taking , in (2.2), we get , a contradiction. This proves that, indeed, F does not satisfy (2.2).
where with .
It is important to note that Opoitsev  was the first who studied coupled fixed points of bivariate mappings (see also [32, 33]) where a double Picard-type iteration sequence of the form (2.4) was used.
In order to state our main results, we need some concepts and results, adapted from the case of mono-variate operators to the case of bivariate operators.
The concept of demicompact operator has been introduced by Petryshyn  (see also  and ) for a mapping , where C is a subset of a Hilbert space H. For the bivariate case it is adapted as follows.
Definition 3 A mapping is called demicompact if it has the property that whenever and are bounded sequences in C with the property that and converge strongly to 0, then there exists a subsequence of such that and strongly.
We need the following version of the well-known Browder-Gohde-Kirk fixed point theorem (see, for example, Theorem 3.1 in ), stated here in the Hilbert space setting.
Theorem 2 Let C be a bounded, closed and convex subset of a Hilbert space H and let be a (weakly) nonexpansive operator. Then F has at least one coupled fixed point in C.
Proof Let be given by , . By the (weakly) nonexpansiveness property of F, we obtain the nonexpansiveness of T and hence, by the Browder-Gohde-Kirk fixed point theorem, it follows that . □
Remark 1 Theorem 2 shows that F has at least one (coupled) fixed point of the form , but, in general, for a bivariate mapping F it is also possible to have coupled fixed points with unequal components, i.e., such that , as shown by the following example.
Then F is weakly Lipschitzian with constants and (in the sense of Definition 1) and F possesses two coupled fixed points , with equal components and two coupled fixed points with unequal components, and .
3 Main results
The main result of this paper is the following strong convergence theorem for a double Krasnoselskij-type algorithm associated with bivariate weakly nonexpansive operators on Hilbert spaces.
where , converges (strongly) to a coupled fixed point of F.
Proof By Theorem 2, F has at least one coupled fixed point with equal components, .
We first show that the sequence converges strongly to zero.
By (3.7), , which shows that is a coupled fixed point of F.
Using now the inequality (3.6), with , we deduce that the sequence of nonnegative real numbers is nonincreasing, hence convergent.
Since its subsequence converges to 0, it follows that the sequence itself converges to 0, that is, the sequence converges strongly to , as . □
Remark 2 Any nonexpansive bivariate mapping is weakly nonexpansive. Hence, by Theorem 3, we obtain Corollary 2.3 in .
We now introduce the concept of demicompactness at a point for a bivariate operator (adapted from the original definition of Petryshyn ).
Definition 4 A map F of into H is said to be demicompact at if, for any bounded sequence in C such that as , there exist a subsequence and an x in C such that as and .
Remark 3 Clearly, if F is demicompact on C, then it is demicompact at 0 but the converse is not true.
The demicompactness of F on the whole C in Theorem 3 may be weakened to the demicompactness at 0, provided that F is continuous.
Theorem 4 Let H be a Hilbert space, C a closed, bounded and convex subset of H, and a weakly nonexpansive mapping such that F is demicompact at 0.
Then the Krasnoselskij-type double sequence given by in C and (3.1) converges (strongly) to a coupled fixed point of F.
Proof Note that in the proof of Theorem 3, we actually used the demicompactness of F at 0, so the arguments used there can be applied here. □
Remark 4 The conclusion of Theorem 4 remains true if instead of the demicompactness of F at 0, we suppose maps closed sets in C into closed sets of H (see ).
If in Theorems 3 and 4, we remove the demicompactness assumption, then (see ), the Krasnoselskij iteration does no longer converge strongly, in general, but it could converge (at least) weakly to a fixed point, as shown in the next theorem, which extends Theorem 3.3 in .
Denote by , the set of all coupled fixed points of F with equal components, i.e., .
converges weakly to p, for any .
Remark 5 The assumption in Theorem 5 may be removed to obtain the following more general result (similar to Theorem 3.4 in ).
converges weakly to a coupled fixed point of F.
is closed, convex, and, hence, weakly compact.
giving a contradiction.
which means that . □
4 Conclusions and further study
Since , it follows that converges to as , for any initial value .
This shows that, for weakly nonexpansive mappings, by using a Krasnoselskij-type iteration we can reach the convergence, while, by means of Picard-type iterations, this cannot be obtained, in general. Indeed, in this case, the Picard-type iteration associated with F is given by , , which is not convergent (except for the case ).
Remark 6 It is important at this stage to say that the coupled fixed point theorems existing in literature, see [1, 4–7, 10, 13, 15–18, 23–31] (only a short list is cited here), cannot be applied to the bivariate functions in Examples 1 and 3.
and it is easily seen that still converges to , the unique coupled fixed point of F, for all .
This also indicates that it is not necessary to consider only the case of a double sequence with equal components in Theorems 3-6 (but the proof of a convergence theorem for such an iterative method will be essentially different from the one given in this paper).
To conclude this paper, we note that, for the general case of a weakly nonexpansive bivariate mapping F, the Picard-type iteration process (2.4) does not generally converge or, even if it converges, its limit is not a coupled fixed point of F, but the Krasnoselskij-type iteration process always converges to a coupled fixed point of F.
In the same way, we can prove convergence theorems for iterative methods of Krasnoselskij type for tripled fixed points, quadruple fixed points etc. of weakly nonexpansive mappings (see [8, 9, 11, 14, 19–21, 38–51], and references therein).
The paper has been finalized during the first author’s visit of Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals. He gratefully thanks the host for kind hospitality and excellent work facilities offered. The first and third authors’ research was supported by the Grants PN-II-RU-TE-2011-3-239 and PN-II-ID-PCE-2011-3-0087 of the Romanian Ministry of Education and Research. The second author is grateful to KACST, Riyad, for supporting research project ARP-32-34. We are grateful to the anonymous referee for the extremely careful reading of the first version of this manuscript and for the corresponding suggestions for improvement, and especially for pointing out a few (non-fatal) errors in the proofs.
- Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65(7):1379–1393. 10.1016/j.na.2005.10.017View ArticleMathSciNetGoogle Scholar
- Opoitsev VI Ekonomiko-Matematicheskaya Biblioteka 31. In Nelineinaya sistemostatika. Nauka, Moscow; 1986. (Nonlinear Systemostatics. Library of Mathematical Economics)Google Scholar
- Abbas M, Khan AR, Nazir T: Coupled common fixed point results in two generalized metric spaces. Appl. Math. Comput. 2011, 217: 6328–6336. 10.1016/j.amc.2011.01.006View ArticleMathSciNetGoogle Scholar
- Aghajani A, Arab R: Fixed points of -contractive mappings in partially ordered b -metric spaces and application to quadratic integral equations. Fixed Point Theory Appl. 2013., 2013: Article ID 245 10.1186/1687-1812-2013-245Google Scholar
- Aghajani A, Abbas M, Kallehbasti EP: Coupled fixed point theorems in partially ordered metric spaces and application. Math. Commun. 2002, 17(2):497–509.Google Scholar
- Amini-Harandi A: Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem. Math. Comput. Model. 2013, 57(9–10):2343–2348. 10.1016/j.mcm.2011.12.006View ArticleGoogle Scholar
- Aydi H, Samet B, Vetro C:Coupled fixed point results in cone metric spaces for -compatible mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 27 10.1186/1687-1812-2011-27Google Scholar
- Berinde V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74(18):7347–7355. 10.1016/j.na.2011.07.053View ArticleMathSciNetGoogle Scholar
- Berinde V: Coupled coincidence point theorems for mixed monotone nonlinear operators. Comput. Math. Appl. 2012, 64(6):1770–1777. 10.1016/j.camwa.2012.02.012View ArticleMathSciNetGoogle Scholar
- Berinde V: Coupled fixed point theorems for φ -contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 2012, 75(6):3218–3228. 10.1016/j.na.2011.12.021View ArticleMathSciNetGoogle Scholar
- Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 4889–4897. 10.1016/j.na.2011.03.032View ArticleMathSciNetGoogle Scholar
- Berinde V, Păcurar M: Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 115Google Scholar
- Berzig M, Samet B: An extension of coupled fixed point’s concept in higher dimension and applications. Comput. Math. Appl. 2012, 63(8):1319–1334. 10.1016/j.camwa.2012.01.018View ArticleMathSciNetGoogle Scholar
- Borcut M, Berinde V: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput. 2012, 218(10):5929–5936. 10.1016/j.amc.2011.11.049View ArticleMathSciNetGoogle Scholar
- Ćirić L, Damjanović B, Jleli M, Samet B: Coupled fixed point theorems for generalized Mizoguchi-Takahashi contractions with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 51 10.1186/1687-1812-2012-51Google Scholar
- Ibn Dehaish BA, Khamsi MA, Khan AR: Mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces. J. Math. Anal. Appl. 2013, 397(2):861–868. 10.1016/j.jmaa.2012.08.013View ArticleMathSciNetGoogle Scholar
- Gu F, Yin Y: A new common coupled fixed point theorem in generalized metric space and applications to integral equations. Fixed Point Theory Appl. 2013., 2013: Article ID 266 10.1186/1687-1812-2013-266Google Scholar
- Hussain N, Salimi P, Al-Mezel S: Coupled fixed point results on quasi-Banach spaces with application to a system of integral equations. Fixed Point Theory Appl. 2013., 2013: Article ID 261 10.1186/1687-1812-2013-261Google Scholar
- Karapinar E, Berinde V: Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces. Banach J. Math. Anal. 2012, 6(1):74–89. 10.15352/bjma/1337014666View ArticleMathSciNetGoogle Scholar
- Khan AR: Common fixed points and solutions of nonlinear functional equations. Fixed Point Theory Appl. 2013., 2013: Article ID 290Google Scholar
- Khan AR, Abbas M, Ali B: Tripled coincidence and common fixed point theorems for hybrid pair of mappings. Creative Math. Inform. 2013, 22(1):53–64.MathSciNetGoogle Scholar
- Olaoluwa H, Olaleru JO, Chang SS: Coupled fixed point theorems for asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 68Google Scholar
- Sabetghadam F, Masiha HP, Sanatpour AH: Some coupled fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 125426Google Scholar
- Samet B, Vetro C, Vetro P: Fixed point theorems for α - ψ -contractive type mappings. Nonlinear Anal. 2012, 75(4):2154–2165. 10.1016/j.na.2011.10.014View ArticleMathSciNetGoogle Scholar
- Sang Y: A class of φ -concave operators and applications. Fixed Point Theory Appl. 2013., 2013: Article ID 274 10.1186/1687-1812-2013-274Google Scholar
- Shatanawi W, Samet B, Abbas M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. 2012, 55(3–4):680–687. 10.1016/j.mcm.2011.08.042View ArticleMathSciNetGoogle Scholar
- Sintunavarat W, Kumam P, Cho YJ: Coupled fixed point theorems for nonlinear contractions without mixed monotone property. Fixed Point Theory Appl. 2012., 2012: Article ID 170 10.1186/1687-1812-2012-170Google Scholar
- Urs C: Coupled fixed point theorems and applications to periodic boundary value problems. Miskolc Math. Notes 2013, 14(1):323–333.MathSciNetGoogle Scholar
- Wu J, Liu Y: Fixed point theorems for monotone operators and applications to nonlinear elliptic problems. Fixed Point Theory Appl. 2013., 2013: Article ID 134 10.1186/1687-1812-2013-134Google Scholar
- Xiao J-Z, Zhu X-H, Shen Z-M: Common coupled fixed point results for hybrid nonlinear contractions in metric spaces. Fixed Point Theory 2013, 14(1):235–249.View ArticleMathSciNetGoogle Scholar
- Alghamdi MA, Hussain N, Salimi P: Fixed point and coupled fixed point theorems on b -metric-like spaces. J. Inequal. Appl. 2013., 2013: Article ID 402 10.1186/1029-242X-2013-402Google Scholar
- Opoitsev VI: Dynamics of collective behavior. III. Heterogenic systems. Autom. Remote Control 1975, 36(1):124–138. (Russian); translated from Avtomat. i Telemeh. 1, 111–124 (1975)MathSciNetGoogle Scholar
- Opoitsev VI, Khurodze TA: Nelineinye operatory v prostranstvakh s konusom. Tbilis. Gos. Univ., Tbilisi; 1984. (Russian) (Nonlinear Operators in Spaces with a Cone)Google Scholar
- Petryshyn WV: Construction of fixed points of demicompact mappings in Hilbert space. J. Math. Anal. Appl. 1966, 14(2):276–284. 10.1016/0022-247X(66)90027-8View ArticleMathSciNetGoogle Scholar
- Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 1967, 20(2):197–228. 10.1016/0022-247X(67)90085-6View ArticleMathSciNetGoogle Scholar
- Petryshyn WV, Williamson TE Jr.: Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings. J. Math. Anal. Appl. 1973, 43(2):459–497. 10.1016/0022-247X(73)90087-5View ArticleMathSciNetGoogle Scholar
- Berinde V: Iterative Approximation of Fixed Points. Springer, Berlin; 2007.MATHGoogle Scholar
- Berinde V, Kovacs G: Stabilizing discrete dynamical systems by monotone Krasnoselskij type iterative schemes. Creative Math. Inform. 2008, 17(3):298–307.MathSciNetGoogle Scholar
- Chidume CE: Geometric Properties of Banach Spaces and Nonlinear Iteration. Springer, Berlin; 2009.MATHGoogle Scholar
- Chidume CE, Măruşter Ş: Iterative methods for the computation of fixed points of demicontractive mappings. J. Comput. Appl. Math. 2010, 234(3):861–882. 10.1016/j.cam.2010.01.050View ArticleMathSciNetGoogle Scholar
- Krasnoselskij MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 1955, 10(1(63)):123–127. (in Russian)Google Scholar
- Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 44: 506–510.View ArticleGoogle Scholar
- Marinescu DŞ, Monea M: About Krasnoselskij iterative method. Creative Math. Inform. 2013, 22(2):199–206.MathSciNetGoogle Scholar
- Păcurar M: Iterative Methods for Fixed Point Approximation. Risoprint, Cluj-Napoca; 2010.MATHGoogle Scholar
- Păcurar M: Common fixed points for almost Presić type operators. Carpath. J. Math. 2012, 28(1):117–126.Google Scholar
- Rafiq A, Acu AM: A new implicit iteration process for two strongly pseudocontractive mappings. Creative Math. Inform. 2012, 21(2):197–201.MathSciNetGoogle Scholar
- Rus IA: Some properties of the solutions of those equations for which the Krasnoselskii iteration converges. Carpath. J. Math. 2012, 28(2):329–336.MathSciNetGoogle Scholar
- Rus IA: An abstract point of view on iterative approximation of fixed points. Fixed Point Theory 2012, 13(1):179–192.MathSciNetGoogle Scholar
- Berinde V: Convergence theorems for fixed point iterative methods defined as admissible perturbations of a nonlinear operator. Carpath. J. Math. 2013, 29(1):9–18.MathSciNetGoogle Scholar
- Berinde V, Khan AR, Păcurar M: Convergence theorems for admissible perturbations of pseudocontractive operators. Miskolc Math. Notes 2014, 15(1):27–37.MathSciNetGoogle Scholar
- Ishikawa S: Fixed points and iterations of a nonexpansive mapping in a Banach space. Proc. Am. Math. Soc. 1976, 59: 65–71. 10.1090/S0002-9939-1976-0412909-XView ArticleMathSciNetGoogle Scholar
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